Metamath Proof Explorer


Theorem lsmdisjr

Description: Disjointness from a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016)

Ref Expression
Hypotheses lsmcntz.p
|- .(+) = ( LSSum ` G )
lsmcntz.s
|- ( ph -> S e. ( SubGrp ` G ) )
lsmcntz.t
|- ( ph -> T e. ( SubGrp ` G ) )
lsmcntz.u
|- ( ph -> U e. ( SubGrp ` G ) )
lsmdisj.o
|- .0. = ( 0g ` G )
lsmdisjr.i
|- ( ph -> ( S i^i ( T .(+) U ) ) = { .0. } )
Assertion lsmdisjr
|- ( ph -> ( ( S i^i T ) = { .0. } /\ ( S i^i U ) = { .0. } ) )

Proof

Step Hyp Ref Expression
1 lsmcntz.p
 |-  .(+) = ( LSSum ` G )
2 lsmcntz.s
 |-  ( ph -> S e. ( SubGrp ` G ) )
3 lsmcntz.t
 |-  ( ph -> T e. ( SubGrp ` G ) )
4 lsmcntz.u
 |-  ( ph -> U e. ( SubGrp ` G ) )
5 lsmdisj.o
 |-  .0. = ( 0g ` G )
6 lsmdisjr.i
 |-  ( ph -> ( S i^i ( T .(+) U ) ) = { .0. } )
7 incom
 |-  ( S i^i ( T .(+) U ) ) = ( ( T .(+) U ) i^i S )
8 7 6 eqtr3id
 |-  ( ph -> ( ( T .(+) U ) i^i S ) = { .0. } )
9 1 3 4 2 5 8 lsmdisj
 |-  ( ph -> ( ( T i^i S ) = { .0. } /\ ( U i^i S ) = { .0. } ) )
10 incom
 |-  ( T i^i S ) = ( S i^i T )
11 10 eqeq1i
 |-  ( ( T i^i S ) = { .0. } <-> ( S i^i T ) = { .0. } )
12 incom
 |-  ( U i^i S ) = ( S i^i U )
13 12 eqeq1i
 |-  ( ( U i^i S ) = { .0. } <-> ( S i^i U ) = { .0. } )
14 11 13 anbi12i
 |-  ( ( ( T i^i S ) = { .0. } /\ ( U i^i S ) = { .0. } ) <-> ( ( S i^i T ) = { .0. } /\ ( S i^i U ) = { .0. } ) )
15 9 14 sylib
 |-  ( ph -> ( ( S i^i T ) = { .0. } /\ ( S i^i U ) = { .0. } ) )